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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.getting_started.overview"></a><a class="link" href="overview.html" title="Overview">Overview</a>
</h3></div></div></div>
<p>
        odeint is a library for solving initial value problems (IVP) of ordinary
        differential equations. Mathematically, these problems are formulated as
        follows:
      </p>
<p>
        <span class="emphasis"><em>x'(t) = f(x,t)</em></span>, <span class="emphasis"><em>x(0) = x0</em></span>.
      </p>
<p>
        <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>f</em></span> can be vectors and the
        solution is some function <span class="emphasis"><em>x(t)</em></span> fulfilling both equations
        above. In the following we will refer to <span class="emphasis"><em>x'(t)</em></span> also
        <code class="computeroutput"><span class="identifier">dxdt</span></code> which is also our notation
        for the derivative in the source code.
      </p>
<p>
        Ordinary differential equations occur nearly everywhere in natural sciences.
        For example, the whole Newtonian mechanics are described by second order
        differential equations. Be sure, you will find them in every discipline.
        They also occur if partial differential equations (PDEs) are discretized.
        Then, a system of coupled ordinary differential occurs, sometimes also referred
        as lattices ODEs.
      </p>
<p>
        Numerical approximations for the solution <span class="emphasis"><em>x(t)</em></span> are calculated
        iteratively. The easiest algorithm is the Euler scheme, where starting at
        <span class="emphasis"><em>x(0)</em></span> one finds <span class="emphasis"><em>x(dt) = x(0) + dt f(x(0),0)</em></span>.
        Now one can use <span class="emphasis"><em>x(dt)</em></span> and obtain <span class="emphasis"><em>x(2dt)</em></span>
        in a similar way and so on. The Euler method is of order 1, that means the
        error at each step is <span class="emphasis"><em>~ dt<sup>2</sup></em></span>. This is, of course, not
        very satisfying, which is why the Euler method is rarely used for real life
        problems and serves just as illustrative example.
      </p>
<p>
        The main focus of odeint is to provide numerical methods implemented in a
        way where the algorithm is completely independent on the data structure used
        to represent the state <span class="emphasis"><em>x</em></span>. In doing so, odeint is applicable
        for a broad variety of situations and it can be used with many other libraries.
        Besides the usual case where the state is defined as a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
        or a <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>, we provide native support for the
        following libraries:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>
          </li>
<li class="listitem">
            <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>, making
            odeint naturally running on CUDA devices
          </li>
<li class="listitem">
            gsl_vector for compatibility with the many numerical function in the
            GSL
          </li>
<li class="listitem">
            <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>
          </li>
<li class="listitem">
            <a href="http://www.boost.org/doc/libs/release/libs/fusion/" target="_top">Boost.Fusion</a>
            (the state type can be a fusion vector)
          </li>
<li class="listitem">
            <a href="http://www.boost.org/doc/libs/release/libs/units/" target="_top">Boost.Units</a>
          </li>
<li class="listitem">
            <a href="http://software.intel.com/en-us/articles/intel-mkl/" target="_top">Intel
            Math Kernel Library</a> for maximum performance
          </li>
<li class="listitem">
            <a href="https://github.com/ddemidov/vexcl" target="_top">VexCL</a> for OpenCL
          </li>
<li class="listitem">
            <a href="http://www.boost.org/doc/libs/release/libs/graph/" target="_top">Boost.Graph</a>
            (still experimentally)
          </li>
</ul></div>
<p>
        In odeint, the following algorithms are implemented:
      </p>
<div class="table">
<a name="boost_numeric_odeint.getting_started.overview.stepper_algorithms"></a><p class="title"><b>Table 1.1. Stepper Algorithms</b></p>
<div class="table-contents"><table class="table" summary="Stepper Algorithms">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Algorithm
                </p>
              </th>
<th>
                <p>
                  Class
                </p>
              </th>
<th>
                <p>
                  Concept
                </p>
              </th>
<th>
                <p>
                  System Concept
                </p>
              </th>
<th>
                <p>
                  Order
                </p>
              </th>
<th>
                <p>
                  Error Estimation
                </p>
              </th>
<th>
                <p>
                  Dense Output
                </p>
              </th>
<th>
                <p>
                  Internal state
                </p>
              </th>
<th>
                <p>
                  Remarks
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Explicit Euler
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">euler</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
                  Output Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Very simple, only for demonstrating purpose
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Modified Midpoint
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">modified_midpoint</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  configurable (2)
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Used in Bulirsch-Stoer implementation
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Runge-Kutta 4
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">runge_kutta4</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  The classical Runge-Kutta scheme, good general scheme without error
                  control
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Cash-Karp
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">runge_kutta_cash_karp54</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  Yes (4)
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Good general scheme with error estimation, to be used in controlled_error_stepper
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Dormand-Prince 5
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  Yes (4)
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Standard method with error control and dense output, to be used
                  in controlled_error_stepper and in dense_output_controlled_explicit_fsal.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Fehlberg 78
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">runge_kutta_fehlberg78</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  Yes (7)
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Good high order method with error estimation, to be used in controlled_error_stepper.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Adams Bashforth
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">adams_bashforth</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  configurable
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Multistep method
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Adams Bashforth Moulton
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">adams_bashforth_moulton</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  configurable
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Combined multistep method
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Controlled Runge-Kutta
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">controlled_runge_kutta</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  depends
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  depends
                </p>
              </td>
<td>
                <p>
                  Error control for <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a>. Requires an <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a> from above. Order depends on the given ErrorStepper
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Dense Output Runge-Kutta
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">dense_output_runge_kutta</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
                  Output Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  depends
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Dense output for <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                  and <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a> from above if they provide dense output functionality
                  (like <code class="computeroutput"><span class="identifier">euler</span></code> and
                  <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>).
                  Order depends on the given stepper.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Bulirsch-Stoer
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">bulirsch_stoer</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  variable
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Stepper with step size and order control. Very good if high precision
                  is required.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Bulirsch-Stoer Dense Output
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">bulirsch_stoer_dense_out</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
                  Output Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/system.html" title="System">System</a>
                </p>
              </td>
<td>
                <p>
                  variable
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Stepper with step size and order control as well as dense output.
                  Very good if high precision and dense output is required.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Implicit Euler
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">implicit_euler</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
                  System</a>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Basic implicit routine. Requires the Jacobian. Works only with
                  <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>
                  vectors as state types.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Rosenbrock 4
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">rosenbrock4</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
                  System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Good for stiff systems. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>
                  vectors as state types.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Controlled Rosenbrock 4
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">rosenbrock4_controller</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
                  Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
                  System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Rosenbrock 4 with error control. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>
                  vectors as state types.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Dense Output Rosenbrock 4
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">rosenbrock4_dense_output</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
                  Output Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/implicit_system.html" title="Implicit System">Implicit
                  System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Controlled Rosenbrock 4 with dense output. Works only with <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>
                  vectors as state types.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Symplectic Euler
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">symplectic_euler</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/symplectic_system.html" title="Symplectic System">Symplectic
                  System</a> <a class="link" href="../concepts/simple_symplectic_system.html" title="Simple Symplectic System">Simple
                  Symplectic System</a>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Basic symplectic solver for separable Hamiltonian system
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Symplectic RKN McLachlan
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/symplectic_system.html" title="Symplectic System">Symplectic
                  System</a> <a class="link" href="../concepts/simple_symplectic_system.html" title="Simple Symplectic System">Simple
                  Symplectic System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Symplectic solver for separable Hamiltonian system with 6 stages
                  and order 4.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Symplectic RKN McLachlan
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_m4_mclachlan</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/symplectic_system.html" title="Symplectic System">Symplectic
                  System</a> <a class="link" href="../concepts/simple_symplectic_system.html" title="Simple Symplectic System">Simple
                  Symplectic System</a>
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Symplectic solver with 5 stages and order 4, can be used with arbitrary
                  precision types.
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Velocity Verlet
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">velocity_verlet</span></code>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a>
                </p>
              </td>
<td>
                <p>
                  <a class="link" href="../concepts/second_order_system.html" title="Second Order System">Second
                  Order System</a>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  No
                </p>
              </td>
<td>
                <p>
                  Yes
                </p>
              </td>
<td>
                <p>
                  Velocity verlet method suitable for molecular dynamics simulation.
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break">
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2009-2015 Karsten Ahnert and Mario Mulansky<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
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